[1]}, [2]} 
Let \(x^* \in \mathbb {R}^n\)  be the solution to the problem (REF ) and denote by \(\eta = y-Ax^*\) .
Given an arbitrary vector \(u \in \mathbb {R}^n\) , let \({\cal Z}^{\#}_k (u)\)  be the optimal \(k\) -thresholding vector of \(u\) .
Then for any \(k\) -sparse binary vector \(w^* \in \lbrace 0,1\rbrace ^n\)  satisfying \(\operatorname{supp}(x^*) \subseteq \operatorname{supp}(w^*)\) , one has
\(*{{\cal Z}^{\#}_k (u) - x^*}_2 \le \sqrt{\frac{1+\delta _k}{1-\delta _{2k}}} *{(x^* - u) \otimes w^*}_2 + \frac{2}{\sqrt{1-\delta _{2k}}} \Vert  \eta \Vert _2.\) 
